Show that if A^{2} is the zero matrix, then the only eigenvalue of A is 0.

Maiclubk

Maiclubk

Answered question

2021-09-13

Show that if A2 is the zero matrix, then the only eigenvalue of A is 0.

Answer & Explanation

yagombyeR

yagombyeR

Skilled2021-09-14Added 92 answers

A2v=A(Av)=A(λv)=λAv=λ2v
Let λ be any eigenvalue of A. Then there exists a nonzero vector v such that Av=λv. Using this the equation on the left implies that A2v=λ2v.
0=λ2vλ=0
Since A2 is the zero matrix, A2v=0 which implies that λ2v=0. Since v is nonzero λ2=0 which implies λ=0. Therefore, λ=0 is the only eigenvalue of A.

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